114 THE THEORY OF SCREWS. [126- 



126. Reaction of Constraints. 



Whatever the constraints may be, their reaction produces an impulsive 

 wrench R l upon the body at the moment of action of the impulsive wrench 

 X^. The two wrenches X l and 7^ compound into a third wrench F,. If 

 the body were free, Y l is the impulsive wrench to which the instantaneous 

 screw A-i would correspond. Since X lt X 2 , X 3 are co-cylindroidal, A lt A z , A 3 

 must be co-cylindroidal, and therefore also must be Y lt F 2 , Y 3 . The nine 

 wrenches X ly X. 2 , X 3 , R lt E, R i} - F,, F 2 , Y 3 must equilibrate; but if 

 Xi, X 2 , X 3 equilibrate, then the twist velocities about A lt A 2 , A a must 

 neutralize, and therefore the wrenches about F, , F 2 , F 3 must equilibrate. 

 Hence RI, R. 2 , R 3 equilibrate, and are therefore co-cylindroidal. 



Following the same line of proof used in the last section, we can show 

 that 



If impulsive wrenches on any four co-cylindroidal screws act upon a 

 partially free rigid body, the four corresponding initial reactions of the 

 constraints also constitute wrenches about four co-cylindroidal screws; and, 

 further, the anharmonic ratios of the two groups of four screws are equal. 



127. Principal Screws of Inertia. 



If a quiescent body with freedom of the second order receive impulsive 

 wrenches on three screws X l} X. 2 , X 3 on the cylindroid which expresses the 

 freedom, and if the corresponding instantaneous screws on the same cylin 

 droid be A lt A%, A s , then the relation between any other impulsive screw X 

 on the cylindroid and the corresponding instantaneous screw A is completely 

 defined by the condition that the anharmonic ratio of X, X lt X 2 , X 3 is equal 

 to the anharmonic ratio of A, A^, A.,, A 3 . 



If three rays parallel to X lt X 2 , X 3 be drawn from a point, and from the 

 same point three rays parallel to A^, A.,, A 3 , then, all six rays being in the 

 same plane, it is well known that the problem to determine a ray Z such 

 that the anharmonic ratio of Z, A l , A 2 , A 3 is equal to that of Z, X^, X 2 , X 3) 

 admits of two solutions. There are, therefore, two screws on a cylindroid 

 such that an impulsive wrench on one of these screws will cause the 

 body to commence to twist about the same screw. 



We have thus arrived by a special process at the two principal screws of 

 inertia possessed by a body which has freedom of the second order. This is, 

 of course, a particular case of the general theorem of 78. We shall show 

 in the next section how these screws can be determined in another manner. 



128. The Ellipse of Inertia. 



We have seen ( 89) that a linear parameter u a may be conceived appro 

 priate to any screw a of a system, so that when the body is twisting about 



